Multiple positive solutions for a class of nonlinear four-point boundary value problem with <Emphasis Type="Italic">p</Emphasis>-Laplacian

This paper deals with the existence of multiple positive solutions for a class of nonlinear singular four-point boundary value problem with p-Laplacian：
{（φ（u′））′＋a（t）f（u（t））=0, 0〈t〈1,
αφ（u（0））-βφ（u′（ξ））=0,γφ（u（1））＋δφ（u′（η））0,
where φ（x） = ｜x｜^p-2x,p 〉 1, a（t） may be singular at t = 0 and/or t = 1. By applying Leggett-Williams fixed point theorem and Schauder fixed point theorem, the sufficient conditions for the existence of multiple （at least three） positive solutions to the above four-point boundary value problem are provided. An example to illustrate the importance of the results obtained is also given.     

ON THE EMDEN-FOWLER EQUATION u″（t）u（t） = c1＋c2u′（t）^2 WITH c1≥0, c2≥0

In this article, we study the following initial value problem for the nonlinear equation
{u″u（t）=c1＋c2u′（t）^2, c1≥0, c2≥0,
u（0）=u0, u′（0）=u1.
We are interested in properties of solutions of the above problem. We find the life-span, blow-up rate, blow-up constant and the regularity, null point, critical point, and asymptotic behavior at infinity of the solutions.     

Multiple positive solutions for superlinear second order singular boundary value problems with derivative dependence

The existence of at least two positive solutions is presented for the singular second-order boundary value problem
{1/p（t）（ p（t）x′（t））′＋Φ（t）f（t,x（t）,p（t）x′（t））=0,0〈t〈1,
limt→0 p（t）x′（t）=0,x（1）=0
by using the fixed point index, where f may be singular at x = 0 and px ′= 0.     

Existence Results for a Class of Semilinear Elliptic Systems

In this paper, we study the existence of nontrivial solutions for the problem
{-△u=f（x,u,v）＋h1（x）in Ω
-△v=g（x,u,v）＋h2（x）inΩ
u=v=0 onδΩ
where Ω is bounded domain in R^N and h1,h2 ∈ L^2 （Ω）. The existence result is obtained by using the Leray-Schauder degree under the following condition on the nonlinearities f and g：
{lim s,｜t｜→＋∞f（x,s,t）/s=lim ｜s｜,t→＋∞g（x,s,t）/t=λ＋1 uniformly on Ω,
lim -s,｜t｜→＋∞f（x,s,t）/s=lim ｜s｜,-t→＋∞g（x,s,t）/t=λ-,uniformly on Ω,
where λ＋,λ-∈（0）∪σ（-△）,σ（-△）denote the spectrum of -△. The cases （i） where λ＋ = λ_ and （ii） where λ＋≠λ_ such that the closed interval with endpoints λ＋,λ_ contains at most one simple eigenvatue of -△ are considered.     

CLASSIFICATION OF POSITIVE SOLUTIONS FOR NONLINEAR DIFFERENTIAL AND INTEGRAL SYSTEMS WITH CRITICAL EXPONENTS

We classify all positive solutions for the following integral system：{ui（x）=∫Rn1/│x-y│^n-α fi（u（y））dy,x∈R^n,i=1,…,m,0〈α〈n,and u（x）=（u1（x）,u2（x）…,um（x））.Here fi（u）, 1 ≤ i ≤m, monotone nondecreasing are real-valued functions of homogeneous degree n＋α/n-α and are monotone nondecreasing with respect to all the independent variables U1, u2, ..., urn.In the special case n ≥ 3 and α = 2. we show that the above system is equivalent to thefollowing elliptic PDE system：This system is closely related to the stationary SchrSdinger system with critical exponents for Bose-Einstein condensate     

Existence of positive solutions for a singular <Emphasis Type="Italic">p</Emphasis>-Laplacian differential equation

In this paper, we are concerned with the existence of positive solutions for a singular p-Laplacian differential equation
（φp（u＇））＇＋β/r φp（u＇）-γ ｜u＇｜^p/u ＋ g（r）=0,0〈r〈1,
subject to the Dirichlet boundary conditions： u（0） = u（1） =0, where φp（s） = ｜sl^P-2s,p 〉 2,β 〉0, γ〉（p-1）/p （β ＋ 1）, and g（r） ∈ C^1 [0, 1] with g（r） 〉 0 for all τ ∈ [0, 1]. We use the method of elliptic regularization, by carrying out two limit processes, to get a positive solution.     

Low Regularity Solution of the Initial-Boundary-Value Problem for the ＂Good＂ Boussinesq Equation on the Half Line

we study an initial-boundary-value problem for the ＂good＂ Boussinesq equation on the half line
{δt^2u-δx^2u＋δx^4u＋δx^2u^2=0,t〉0,x〉0.
u（0,t）=h1（t）,δx^2u（0,t） =δth2（t）,
u（x,0）=f（x）,δtu（x,0）=δxh（x）.
The existence and uniqueness of low reguality solution to the initial-boundary-value problem is proved when the initial-boundary data （f, h, h1, h2） belong to the product space
H^5（R^＋）×H^s-1（R^＋）×H^s/2＋1/4（R^＋）×H^s/2＋1/4（R^＋）
1 The analyticity of the solution mapping between the initial-boundary-data and the with 0 ≤ s 〈 1/2. solution space is also considered.     

On the weighted elliptic problems involving multi-singular potentials and multi-critical exponents

Suppose Ω belong to R^N（N≥3） is a smooth bounded domain,ξi∈Ω,0〈ai〈√μ,μ：=（（N-1）/2）^2,0≤μi〈（√μ-ai）^2,ai〈bi〈ai＋1 and pi：=2N/N-2（1＋ai-bi）are the weighted critical Hardy-Sobolev exponents, i = 1, 2,..., k, k ≥ 2. We deal with the conditions that ensure the existence of positive solutions to the multi-singular and multi-critical elliptic problem ∑i=1^k（-div（｜x-ξi｜^-2ai△↓u）-μiu/｜x-ξi｜^2（1＋ai）-u^pi-1/｜x-ξi｜^bipi）=0with Dirichlet boundary condition, which involves the weighted Hardy inequality and the weighted Hardy-Sobolev inequality. The results depend crucially on the parameters ai, bi and #i, i -- 1, 2,..., k.     

Positive Solution of Multi-Point Boundary Value Problems for the One-Dimensional p-Laplacian with Singularities

In this paper, we obtain positive solution to the following multi-point singular boundary value problem with p-Laplacian operator,{（ φp（u＇））＇＋q（t）f（t,u,u＇）=0,0〈t〈1,u（0）=∑i=1^nαiu（ξi）,u＇（1）=∑i=1^nβiu＇（ξi）,whereφp（s）=｜s｜^p-2s,p≥2;ξi∈（0,1）（i=1,2,…,n）,0≤αi,βi〈1（i=1,2,…n）,0≤∑i=1^nαi,∑i=1^nβi〈1,and q（t） may be singular at t=0,1,f（t,u,u＇）may be singular at u＇=0     

Monotone positive solution for three-point boundary value problem

In this paper, the existence of monotone positive solution for the following secondorder three-point boundary value problem is studied：
x″（t）＋f（t,x（t））=0,0〈t〈1,
x′（0）=0,x（1）＋δx′（η）=0,
where η ∈ （0, 1）, δ∈ [0, ∞）, f ∈ C（[0, 1] × [0, ∞）, [0, ∞））. Under certain growth conditions on the nonlinear term f and by using a fixed point theorem of cone expansion and compression of functional type due to Avery, Anderson and Krueger, sufficient conditions for the existence of monotone positive solution are obtained and the bounds of solution are given. At last, an example is given to illustrate the result of the paper.     

一类n-阶m-点奇异边值问题的正解

研究n-阶m-点奇异边值问题其中h(t)允许在t=0,t=1处奇异,f(t,v_0,v_1,…,v_(n-2))允许在v_i=0(i=0,1,…,n-2)处奇异.利用锥拉伸与压缩不动点定理得到了上述奇异边值问题正解的存在性.     

一类带p-Laplacian的Sturm-Liouville型2m点边值问题三个正解的存在性

该文考虑了下面的具一维$p$\,-Laplacian算子的多点边值问题 $ \left\{ \begin{array}{rl} &;\disp (\phi_{p}(x'(t)))'+h(t)f(t,x(t),x'(t))=0,\hspace{3mm}01,~\alpha_{i}>0,~\beta_{i}>0,~0<\sum\limits_{i=1}^{m-1}\alpha_{i}\xi_{i}\leq1,~ 0<\sum\limits_{i=1}^{m-1}\beta_{i}(1-\eta_{i})\leq1,~0=\xi_{0} <\xi_{1}<\xi_{2}<\cdots<\xi_{m-1}<\eta_{1}<\eta_{2}<\cdots<\eta_{m-1}<\eta_{m}=1,~i=1,2,\cdots,m-1.$ 通过运用锥上的不动点定理, 该文得到了至少三个正解的存在性. 有趣的是文中的边界条件是一个新型的Sturm-Liouville型边界条件, 这类边值问题到目前为止还很少被研究.     

二阶微分方程多点边值问题的三重正解

We consider the second-order differential equation u(t) + q(t)f(t,u(t),u (t)) = 0,0 t 1,subject to three-point boundry condition u(0) = 0,u(1) = a 0 u(ξ 0 ),or to m-point boundary conditionu (0) = m2 i=1 b i u (ξ i ),u(1) = m2 i=1 a i u(ξ i ).We show the existence of at least three positive solutions of the above multi-point boundary-value problem by applying a new fixed-point theorem introduced by Avery and Peterson.     

Existence Theory for an Arbitrary Order Fractional Differential Equation with Deviating Argument

In this paper, we are concerned with the existence criteria for positive solutions of the following nonlinear arbitrary order fractional differential equations with deviating argument

具$p$-Laplacian 算子的多点边值问题迭代解的存在性

利用单调迭代技巧和推广的Mawhin定理得到下述带有p-Laplacian算子的多点边值问题迭代解的存在性,{(Фp(u'))' f(t,u, Tu)=0, 0(≤)t(≤)1,u(0)=q-1∑i=1γiu(δi),u(1)=m-1∑i=1ηiu(ξi),其中Фp(s)=|s|p-2s,p＞1;0＜δi＜1,γi＞0,1(≤)i(≤)q-1;0＜ξi＜1,ηi(≥)0,1(≤)i(≤)m-1且q-1∑i=1γi＜1,m-1∑i=1ηi(≤)1;Tu(t)=∫t0k(t,s)u(s)ds,k(t,s)∈C(I×I,R ).     

具$p$-Laplace 算子的三点边值问题正解的存在性

本文用锥上的Krasnoselskii’s不动点定理研究了具有p-Laplace算子的三点边值问题:其中0<α,β<1,0<η<1且(?)p(z)=|x|p-2z,p>1．在f满足一定的增长条件下,得到方程正解的存在性．作为应用,给出两个例子．     

Solvability for Some Higher Order Multi-Point Boundary Value Problems at Resonance

In this paper, by using the Mawhin’s continuation theorem, we obtain an existence theorem for some higher order multi-point boundary value problems at resonance in the following form: $$\begin{array}{lll}x^{(n)}(t) = f(t,x(t),x'(t),\ldots,x^{(n-1)}(t))+e(t),\ t\in(0,1),\\x^{(i)}(0) = 0, i=0,1,\ldots,n-1,\ i\neq p, \\x^{(k)}(1) = \sum\limits_{j=1}^{m-2}{\beta_j}x^{(k)}(\eta_j),\end{array}$$ where ${f:[0,1]\times \mathbb{R}^n \to \mathbb{R}=(-\infty,+\infty)}$ is a continuous function, ${e(t)\in L^1[0,1], p, k\in\{0,1,\ldots,n-1\}}$ are fixed, m ≥ 3 for p ≤ k (m ≥ 4 for p > k), ${\beta_j \in \mathbb{R}, j=1,2,\ldots,m-2, 0 < \eta_1 < \eta_2 < \cdots < \eta_{m-2} <1 }$ . We give an example to demonstrate our results.     

Existence of positive solutions for fractional differential systems with multi point boundary conditions

In this paper, we study the existence of positive solutions to the boundary value problem for the fractional differential system $$\left\{\begin{array}{lll} D_{0^+}^\beta \phi_p(D_{0^+}^\alpha u) (t) = f_1 (t, u (t), v (t)),\quad t \in (0, 1),\\ D_{0^+}^\beta \phi_p(D_{0^+}^\alpha v) (t) = f_2 (t, u (t), v(t)), \quad t \in (0, 1),\\ D_{0^+}^\alpha u(0)= D_{0^+}^\alpha u(1)=0,\; u (0) = 0, \quad u (1)-\Sigma_{i=1}^{m-2} a_{1i}\;u(\xi_{1i})=\lambda_1,\\ D_{0^+}^\alpha v(0)= D_{0^+}^\alpha v(1)=0,\; v (0) = 0, \quad v (1)-\Sigma_{i=1}^{m-2} a_{2i}\; v(\xi_{2i})=\lambda_2, \end{array}\right. $$ where ${1<\alpha,\beta\leq 2, 2 <\alpha + \beta\leq 4, D_{0^+}^\alpha}$ is the Riemann–Liouville fractional derivative of order α. By using the Leggett–Williams fixed point theorem in a cone, the existence of three positive solutions for nonlinear singular boundary value problems is obtained.     

Unbounded positive solutions for m-point time-scale boundary value problems on infinite intervals

In this work, we consider the following second-order m-point boundary value problem on time scales $$\left\{\begin{array}{@{}l}(\phi_{p}(u^{\triangle}(t)))^{\nabla}+h(t)f(t,u(t),u^{\triangle }(t))=0,\quad t\in(0,+\infty)_{\mathbb{T}},\\[4pt]\displaystyle u(0)=\sum_{i=1}^{m-2}\alpha_{i}u(\eta_{i}),\qquad u^{\triangle}(+\infty)=\sum_{i=1}^{m-2}\beta_{i}u^{\triangle}(\eta_{i}).\end{array}\right.$$ We establish new criteria for the existence of at least three unbounded positive solutions. Our results are new even for the corresponding differential $({\mathbb{T}}={\mathbb{R}})$ , difference equation $({\mathbb{T}}={\mathbb{Z}})$ and for the general time-scale setting. An example is given to illustrate our results.     

Some necessary and sufficient conditions for existence of positive solutions for third order singular super-linear multi-point boundary value problems

We mainly study the existence of positive solutions for the following third order singular super-linear multi-point boundary value problem $$ \left \{ \begin{array}{l} x^{(3)}(t)+ f(t, x(t), x'(t))=0,\quad0 where \(0\leq\alpha_{i}\leq\sum_{i=1}^{m_{1}}\alpha_{i}<1\) , i=1,2,…,m ₁, \(0<\xi_{1}< \xi_{2}< \cdots<\xi_{m_{1}}<1\) , \(0\leq\beta_{j}\leq\sum_{i=1}^{m_{2}}\beta_{i}<1\) , j=1,2,…,m ₂, \(0<\eta_{1}< \eta_{2}< \cdots<\eta_{m_{2}}<1\) . And we obtain some necessary and sufficient conditions for the existence of C ¹[0,1] and C ²[0,1] positive solutions by means of the fixed point theorems on a special cone. Our nonlinearity f(t,x,y) may be singular at t=0 and t=1.